Angles Formed by Transversals Lesson 3.3

Objectives Identify angles formed by transversals.

Key Vocabulary Transversal Interior angles Exterior angles Corresponding angles Alternate interior angles Alternate exterior angles Same-side interior angles

Transversal Transversal - a line that intersects two or more coplanar lines at different points. Line ℓ is a transversal of lines q and r. Line ℓ forms a total of eight angles with lines q and r. These angles, and specific pairings of these angles, are given special names. 2 4 7 1 3 5 6 8 Two coplanar lines Transversal ℓ q r

Transversal Lines intersected by a transversal can be parallel or not parallel.

Transversal Angle Pair Relationships 2 4 7 1 3 5 6 8 ℓ q r exterior interior Interior angles (4 ∠’s) lie in the region between lines q and r. (∠3, ∠4, ∠5, ∠6) Exterior angles (4 ∠’s) lie in the two regions that are not between lines q and r, the regions above line q and below line r. (∠1, ∠2, ∠7, ∠8)

Transversal Angle Pair Relationships 2 4 7 1 3 5 6 8 ℓ q r exterior interior Same-side interior angles – pair of interior angles that lie on the same side of transversal ℓ. (∠3 & ∠5, ∠4 & ∠6) Alternate interior angles – pair of nonadjacent interior angles that lie on opposite sides of transversal ℓ. (∠3 & ∠6, ∠4 & ∠5)

Transversal Angle Pair Relationships 2 4 7 1 3 5 6 8 ℓ q r exterior interior Alternate exterior angles – pair of nonadjacent exterior angles that lie on opposite sides of transversal ℓ. (∠1 & ∠8, ∠2 & ∠7) Corresponding angles – pair of angles that lie on the same side of transversal ℓ and on the same side of lines q and r. (∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8)

Check For Understanding

Pairs Of Angles Formed by a Transversal A line that intersects two or more lines at different points is called a transversal. G F Line L (transversal) Line M B A P Line N D C Q Four angles are formed at point P and another four at point Q by the transversal L. Line M and line N are parallel lines. Line L intersects line M and line N at point P and Q. Eight angles are formed in all by the transversal L.

Corresponding Angles A transversal creates two groups of four angles in each group. Corresponding angles are two angles, one in each group, in the same relative position. 1 2 m 3 4 5 6 n 7 8

Alternate Interior Angles When a transversal cuts two lines, alternate interior angles are angles within the two lines on alternate sides of the transversal. m 1 3 4 2 n

Alternate Exterior Angles When a transversal cuts two lines, alternate exterior angles are angles outside of the two lines on alternate sides of the transversal. 1 3 m n 4 2

Same-Side Interior Angles When a transversal cuts two lines, interior angles on the same side of the transversal are angles within the two lines on the same side of the transversal. m 1 3 2 4 n

Name the pairs of the following angles formed by a transversal. Line M B A Line N D E P Q G F Line L Line M B A Line N D E P Q G F Line L Line M B A Line N D E P Q G F Line L Test Yourself Corresponding angles Alternate interior angles Same-side interior angles

Parallel Lines and Transversals Determine if the statement is true or false. If false, correct the statement. 1. Line r is a transversal of lines p and q. True – Line r intersects both lines in a plane. 2. ∠2 and ∠6 are alternate interior angles. False - The angles are corresponding angles on transversal p. r 16 15 14 13 12 11 10 9 8 7 6 5 3 4 2 1 s p q

Parallel Lines and Transversals Determine if the statement is true or false. If false, correct the statement. 3. ∠9 and ∠11 are alternate interior angles. False – The angles are vertical angles created by the intersection of q and r. 4. ∠1 and ∠7 are alternate exterior angles. True - The angles are alternate exterior angles on transversal p. r 16 15 14 13 12 11 10 9 8 7 6 5 3 4 2 1 s p q

Parallel Lines and Transversals Determine if the statement is true or false. If false, correct the statement. 5. ∠12 and ∠14 are alternate interior angles. True – The angles are alternate interior angles on transversal q. 6. ∠6 and ∠13 are same-side interior angles. True – The angles are same-side interior angles on transversal s. r 16 15 14 13 12 11 10 9 8 7 6 5 3 4 2 1 s p q

Parallel Lines and Transversals Determine if the statement is true or false. If false, correct the statement. 7. ∠9 and ∠10 are alternate exterior angles. False – The angles are a linear pair with linear rays on line r. 8. ∠8 and ∠16 are corresponding angles. True – The angles are corresponding on transversal s. r 16 15 14 13 12 11 10 9 8 7 6 5 3 4 2 1 s p q

Example 1 Describe the relationship between the angles. a. 1 and 2 SOLUTION alternate interior angles a. alternate exterior angles b. same-side interior angles c. 20

Example 2 List all pairs of angles that fit the description. a. corresponding b. alternate exterior c. alternate interior d. same-side interior SOLUTION corresponding a. 1 and 5 2 and 6 3 and 7 4 and 8 21

Example 2 alternate exterior: b. 1 and 8 3 and 6 alternate interior: c. 2 and 7 4 and 5 same-side interior: d. 2 and 5 4 and 7 22

Your Turn: Describe the relationship between the angles. 2 and 7 1. ANSWER alternate exterior angles 3 and 5 2. ANSWER same-side interior angles 1 and 5 3. ANSWER corresponding angles

4 and 5 4. ANSWER alternate interior angles 4 and 8 5. ANSWER corresponding angles 4 and 6 6. ANSWER same-side interior angles

Example 3a: Identify ∠7 and ∠3 as alternate interior, alternate exterior, corresponding, or same-side interior angles. Answer: corresponding

Example 3b: Identify ∠8 and ∠2 as alternate interior, alternate exterior, corresponding, or same-side interior angles. Answer: alternate exterior

Example 3c: Identify ∠5 and ∠1 as alternate interior, alternate exterior, corresponding, or same-side interior angles. Answer: corresponding

Example 3d: Identify ∠7 and ∠1 as alternate interior, alternate exterior, corresponding, or same-side interior angles. Answer: alternate exterior

Example 3e: Identify ∠3 and ∠9 as alternate interior, alternate exterior, corresponding, or same-side interior angles. Answer: alternate interior

Example 3f: Identify ∠7 and ∠10 as alternate interior, alternate exterior, corresponding, or same-side interior angles. Answer: same-side interior

Your Turn: Identify each pair of angles as alternate interior, alternate exterior, corresponding, or same-side interior angles. a. b. c. Answer: same-side interior Answer: corresponding Answer: alternate exterior

Your Turn: Identify each pair of angles as alternate interior, alternate exterior, corresponding, or same-side interior angles. d. e. f. Answer: alternate interior Answer: corresponding Answer: alternate exterior

To determine which line is the transversal for a given angle pair, locate the line that connects the vertices. Helpful Hint

Example 4: Identifying Angle Pairs and Transversals Identify the transversal and classify each angle pair. A. 1 and 3 transversal l corr. s B. 2 and 6 transversal n alt. int s C. 4 and 6 transversal m alt. ext s

Your Turn Identify the transversal and classify the angle pair 2 and 5 in the diagram. transversal n consecutive int. s.

Joke Time What has 18 legs and catches flies? A baseball team. Who stole the soap? The robber ducky! What happened to the plant on the windowsill of the classroom? It grew square roots!

Assignment Section 3.3, pg. 123-125: #1-8 all, 9-41 odd